messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation -2e^(1(-5((3x)/(2x + 3)) - 2)) + 1 = 3sin(1π((3x + 3)/(-1x)) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "How many solutions does the equation 1(5(-2sin(3πx) - 2) - 1)^2 + 3(5(-2sin(3πx) - 2) - 1) - 5 = -2sin(1π(5x^2 + 5x - 2) - 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(2((-2x + 3)/(-3x + 1)) + 1)^2 - 4(2((-2x + 3)/(-3x + 1)) + 1) + 2 and g(x) = -3cos(2π(3x^2 - 4x + 5) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation 1((3(3|(-1x^2 - 3x + 2) + 1|) - 3)/(1(3|(-1x^2 - 3x + 2) + 1|) + 1)) - 1 = 3|((-2(-3sin(2πx - 3) - 1) - 2)/(-1(-3sin(2πx - 3) - 1) - 3)) + 1| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-1|(4(-1x + 1)^2 + 3(-1x + 1) + 2)| - 2) + 1)/(-2(-1|(4(-1x + 1)^2 + 3(-1x + 1) + 2)| - 2) + 1) and g(x) = 3sin(1π(-1(-2|x - 1| + 1) + 3) - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2cos(2π(3|(3x^2 + 1) - 2| + 1) + 1) + 2 and g(x) = 1e^(2((1x - 1)/(-2x - 3))) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1594 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(5(3cos(2π((3x + 2)/(2x - 1)) + 2) + 2) + 4)^2 + 3 and g(x) = 2|(5((1x + 3)/(2x - 1))^2 - 1((1x + 3)/(2x - 1)) - 3)| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 42 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(3sin(1π(2x^2 - 1x + 1) - 1) + 1) and g(x) = 5(-3sin(3πx + 3) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 89 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-2cos(1π((1x + 1)/(1x + 2)) - 3) - 1)^2 + 4(-2cos(1π((1x + 1)/(1x + 2)) - 3) - 1) + 5 and g(x) = 3(2e^(2x) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 37 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(2(-2cos(2πx - 3)) + 2) - 3| - 2 and g(x) = (3(2x^2 + 2x + 5) - 3)/(-3(2x^2 + 2x + 5) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-1sin(3π(-3|(4x^2 - 3x + 3)| - 2) + 3) + 3) - 2)/(-1(-1sin(3π(-3|(4x^2 - 3x + 3)| - 2) + 3) + 3) + 2) and g(x) = -2cos(3π(1(-3x + 3)^2 - 1) - 3) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your a... | 1541 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-2(1cos(3πx - 2) - 1) - 3) - 1)/(-3(-2(1cos(3πx - 2) - 1) - 3) + 2) and g(x) = (1(3|x - 3| - 1) + 2)/(1(3|x - 3| - 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(3π(-3|((-2x)/(-2x - 2))|)) and g(x) = (-1(-2x + 4) - 1)/(1(-2x + 4) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 64 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(-3sin(2π(-1((-3x + 1)/(2x - 3))^2 - 3((-3x + 1)/(2x - 3)) - 4)) - 2) - 1 and g(x) = -3sin(1π(2|(2x^2 + 2x + 2) - 2| + 1) + 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 134 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(4((-2x - 1)/(1x - 2)) + 1)^2 + 4(4((-2x - 1)/(1x - 2)) + 1) - 2 and g(x) = 4(-3cos(2πx) - 2)^2 + 5(-3cos(2πx) - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many solutions does the equation 2sin(3π(2(2|x - 1| - 1)^2 - 3(2|x - 1| - 1) - 4) - 3) - 1 = 1((2x - 1)/(-1x - 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 994 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-1|(5((-2x)/(3x + 2)) + 1) + 3| + 2)^2 + 4(-1|(5((-2x)/(3x + 2)) + 1) + 3| + 2) - 2 and g(x) = 3|((2(2sin(3πx + 2) - 2) - 1)/(-3(2sin(3πx + 2) - 2) - 2)) - 2| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format:... | 45 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1((-1(5(3|x + 1| - 1)^2 + 3(3|x + 1| - 1) - 3))/(1(5(3|x + 1| - 1)^2 + 3(3|x + 1| - 1) - 3) - 2)) + 1 and g(x) = (-1(4(-3|x - 3| + 2)^2 + 4(-3|x - 3| + 2) - 5))/(1(4(-3|x - 3| + 2)^2 + 4(-3|x - 3| + 2) - 5) + 3) in the range -10 ≤... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3cos(2π(5(-1|x - 1| - 2) - 1) + 1) + 3 and g(x) = (1(-3cos(3πx + 2) - 1))/(-2(-3cos(3πx + 2) - 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(5((-2(2sin(3πx + 2) + 1) - 2)/(-3(2sin(3πx + 2) + 1) - 3)) + 2)^2 + 3(5((-2(2sin(3πx + 2) + 1) - 2)/(-3(2sin(3πx + 2) + 1) - 3)) + 2) - 5 and g(x) = (1(5(-1cos(1πx + 2) + 1) - 5) - 2)/(-3(5(-1cos(1πx + 2) + 1) - 5) - 1) wh... | 6 | MATH |
[
{
"content": "How many solutions does the equation (-3(1|(5(3x^2 - 1x) - 5) + 1| - 1) - 3)/(-3(1|(5(3x^2 - 1x) - 5) + 1| - 1) - 2) = -2sin(3π(1|(-1x^2 - 5x + 1) + 3| - 3) + 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 94 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(1cos(1π(-3x^2 - 2x - 5) - 3)) + 2| + 1 and g(x) = 1e^(-1(-1|x - 1| + 1)) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-3cos(3π(5(3|x| - 3)^2 - 2(3|x| - 3) + 5) + 1) + 3) - 2)/(2(-3cos(3π(5(3|x| - 3)^2 - 2(3|x| - 3) + 5) + 1) + 3) - 1) and g(x) = 1((1(3sin(3πx - 2) - 3) - 2)/(1(3sin(3πx - 2) - 3) + 2))^2 - 4((1(3sin(3πx - 2) - 3) - 2)/(1(3... | 114 | MATH |
[
{
"content": "How many solutions does the equation -3|(3cos(1π(5x^2 - 1x - 1) - 1))| - 2 = -3|(4x^2 + 4x) + 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(1(-3(-1|x - 2| + 2))^2 - 5(-3(-1|x - 2| + 2)) + 4) + 1)/(-2(1(-3(-1|x - 2| + 2))^2 - 5(-3(-1|x - 2| + 2)) + 4) - 3) and g(x) = -2cos(2π(2|((-2x - 3)/(2x - 3)) + 2|) - 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in La... | 100 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(3(3x^2 + 1x + 5) - 3) + 3)/(2(3(3x^2 + 1x + 5) - 3) + 1) and g(x) = 3cos(1π(1|x| + 1) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5((-1(3sin(3πx - 2) - 2) - 1)/(3(3sin(3πx - 2) - 2) - 1)) - 5 and g(x) = -1(3sin(2πx - 3) + 1) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(3π(5(-1(-1|x + 3| + 1)^2 + 1(-1|x + 3| + 1) - 5)) + 2) + 2 and g(x) = -4(5(-3|x - 1| + 3))^2 - 1(5(-3|x - 1| + 3)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4((-2(3x^2 - 4) + 3)/(1(3x^2 - 4) + 3)) - 2 and g(x) = 2|(-2cos(1πx) + 3) - 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "How many solutions does the equation -1sin(3π(1((-2x)/(-3x + 1))^2 + 5((-2x)/(-3x + 1))) + 1) + 2 = -2(-3cos(1πx) + 1)^2 - 2(-3cos(1πx) + 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 51 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3cos(1π(-3((2x + 2)/(-2x + 3))^2 - 4((2x + 2)/(-2x + 3)) + 2) - 1) + 2 and g(x) = (3(3x^2 - 1x - 5) + 3)/(-3(3x^2 - 1x - 5) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 41 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(5(3cos(1πx + 3) + 3) + 1) + 1| + 1 and g(x) = 3(-1x + 2)^2 + 2(-1x + 2) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(1|((-1x - 1)/(2x - 3)) + 2| + 1)^2 + 1(1|((-1x - 1)/(2x - 3)) + 2| + 1) + 4 and g(x) = -1(-1sin(1πx) - 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation 1|(-1(5x - 3)^2 + 3(5x - 3) + 1) + 2| - 2 = 3cos(3π(5x - 5) - 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-2sin(1π((-1(1x - 2) - 3)/(-3(1x - 2) - 2)) + 1))^2 + 4(-2sin(1π((-1(1x - 2) - 3)/(-3(1x - 2) - 2)) + 1)) - 5 and g(x) = (1(1cos(2π(1x^2 + 5x - 2) - 3) - 2))/(2(1cos(2π(1x^2 + 5x - 2) - 3) - 2)) intersect in the interval [-10, 10].\n\nPresent the ans... | 47 | MATH |
[
{
"content": "How many solutions does the equation -2((1(3|(1cos(3πx) - 1) + 1|) - 2)/(2(3|(1cos(3πx) - 1) + 1|) - 3))^2 - 4((1(3|(1cos(3πx) - 1) + 1|) - 2)/(2(3|(1cos(3πx) - 1) + 1|) - 3)) + 5 = 2cos(2π(5((1x + 3)/(-3x - 2)) - 3) - 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex for... | 216 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(2sin(3π(-3(-1|x - 1| + 1)^2 - 4(-1|x - 1| + 1) + 5)) - 1) - 5 and g(x) = -1sin(2π((1(-2|x - 1| + 1))/(2(-2|x - 1| + 1) + 2)) + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 411 | MATH |
[
{
"content": "How many solutions does the equation -4(-3e^(1(-4((-2x - 1)/(3x + 2))^2 + 5((-2x - 1)/(3x + 2)) + 1))) + 5 = (-1(-2cos(1π(1x - 1) + 3) - 1) - 3)/(-2(-2cos(1π(1x - 1) + 3) - 1) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(2|(1(1x + 2)^2 - 3(1x + 2) - 1) + 3| - 2) - 2)/(-1(2|(1(1x + 2)^2 - 3(1x + 2) - 1) + 3| - 2) - 3) and g(x) = 5(-1sin(1π(-5x^2 + 1x - 4) + 2) + 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 769 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-2|(-1sin(1πx + 3) - 3) - 3| + 2) - 2)/(-3(-2|(-1sin(1πx + 3) - 3) - 3| + 2) - 2) and g(x) = -2|(3cos(2πx + 2) + 3) - 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(5(1sin(2π((-2x - 1)/(-3x + 2)) + 3) - 2) - 2)^2 - 4(5(1sin(2π((-2x - 1)/(-3x + 2)) + 3) - 2) - 2) + 2 and g(x) = -3|(-3(2sin(3πx - 2) + 3)^2 + 3(2sin(3πx - 2) + 3) + 1) - 2| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex for... | 42 | MATH |
[
{
"content": "How many solutions does the equation -3e^(2(-2|(1x^2 - 4) + 3| - 3)) - 2 = -3cos(2π(3x + 2) + 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 50 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(1((2x)/(1x)))^2 + 2(1((2x)/(1x))) - 1 and g(x) = -4(2cos(2πx - 1) - 1)^2 + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4((3(1sin(2π(-3x^2 + 2) + 2)) + 2)/(-2(1sin(2π(-3x^2 + 2) + 2)) - 3)) - 4 and g(x) = -3(-3(2sin(2πx - 1) + 1) + 4)^2 + 2(-3(2sin(2πx - 1) + 1) + 4) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your ... | 132 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|((1(-5(2x - 4)^2 - 2(2x - 4) + 2) - 1)/(-3(-5(2x - 4)^2 - 2(2x - 4) + 2) - 1)) - 1| and g(x) = -2cos(2π(-2(-2|x + 3|) - 3) - 3) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 160 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(-3(2cos(2πx - 2) + 1) + 4) + 3)/(-3(-3(2cos(2πx - 2) + 1) + 4)) and g(x) = 1(1x^2 + 2x) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many solutions does the equation -1(-1|((1(-1x^2 + 1x + 2) - 2)/(1(-1x^2 + 1x + 2) + 2)) + 1| - 1) + 4 = 5(-2((2x + 1)/(3x)) - 3)^2 + 2(-2((2x + 1)/(3x)) - 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(-2e^(2(-4(-3|x + 1| - 1)^2 - 1(-3|x + 1| - 1) + 1)) - 2) - 3 and g(x) = -3|(-3sin(2π(1x^2 - 5x) + 2) - 3)| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 424 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(-4((1x)/(2x + 2))^2 + 1((1x)/(2x + 2)) + 2) and g(x) = 3|(-2cos(2πx + 2)) + 3| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "How many solutions does the equation 5(4(-3sin(2πx - 2))^2 + 3(-3sin(2πx - 2)) + 2) + 3 = -3(-2|x - 2| - 1) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 64 | MATH |
[
{
"content": "How many solutions does the equation -2cos(3π(5(4x - 1)^2 + 5) + 3) + 2 = -1(-2|x - 1| - 1)^2 + 1(-2|x - 1| - 1) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 54 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(2|(2(2cos(1πx + 3) - 1)^2 + 2(2cos(1πx + 3) - 1) + 4) - 2| + 1) - 3)/(2(2|(2(2cos(1πx + 3) - 1)^2 + 2(2cos(1πx + 3) - 1) + 4) - 2| + 1) - 1) and g(x) = -2|(-2sin(3π(4x + 1) + 3) + 2) - 1| + 3 intersect in the interval [-10, 10].\n\nPresent the answ... | 240 | MATH |
[
{
"content": "How many solutions does the equation -2|(-2cos(3π((1x + 2)/(2x - 1)) + 2) + 2) - 2| + 2 = -3sin(3π(5x^2 + 5x - 4) - 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1438 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(-1|((1x)/(-1x - 1)) - 1| + 2)^2 - 4(-1|((1x)/(-1x - 1)) - 1| + 2) - 5 and g(x) = 1|(-1sin(3πx - 3)) - 1| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation 2sin(3π(-1(1((-1x + 2)/(3x + 1)))^2 - 5(1((-1x + 2)/(3x + 1))) + 1) - 3) - 2 = -3|((-1(-1x - 1) + 1)/(-1(-1x - 1) - 2)) + 3| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((-1(-3(-3cos(2πx + 3) - 2) - 5) + 3)/(2(-3(-3cos(2πx + 3) - 2) - 5) - 1))^2 + 5((-1(-3(-3cos(2πx + 3) - 2) - 5) + 3)/(2(-3(-3cos(2πx + 3) - 2) - 5) - 1)) - 2 and g(x) = 4(1e^(1(-2|x - 3| - 3)))^2 - 2(1e^(1(-2|x - 3| - 3))) ... | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-3cos(3π((-3x - 1)/(-1x - 3)) - 3) - 1) - 2| + 3 and g(x) = -3cos(3π((1x - 1)/(-1x + 2)) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 106 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(2cos(2π((-3(-1x - 3) + 2)/(-1(-1x - 3) - 3))))^2 - 5 and g(x) = -3e^(1(-1(-2|x - 2| + 2) + 1)) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 47 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5((1(3cos(3π(-1x - 2) + 2) - 2) - 2)/(1(3cos(3π(-1x - 2) + 2) - 2) + 2))^2 + 3((1(3cos(3π(-1x - 2) + 2) - 2) - 2)/(1(3cos(3π(-1x - 2) + 2) - 2) + 2)) + 4 and g(x) = 1((2(3x^2 + 3x - 4) + 2)/(3(3x^2 + 3x - 4) + 3)) + 5 in the range ... | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3cos(3π(1((3(-3x + 1) - 1)/(-3(-3x + 1)))^2 - 1) + 3) - 1 and g(x) = -1(5((-1x - 1)/(1x + 3)) + 2)^2 + 3(5((-1x - 1)/(1x + 3)) + 2) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 30 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(3cos(2π(2|x + 3|) - 1) - 3) - 1)/(-1(3cos(2π(2|x + 3|) - 1) - 3)) and g(x) = 2cos(3π(3x^2 + 4x - 1) + 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 93 | MATH |
[
{
"content": "How many solutions does the equation -3(-1|(5(-3sin(1πx - 1))^2 - 1(-3sin(1πx - 1)) - 4) - 3| + 1) = (-2(-1(4x + 5)^2 + 4(4x + 5) + 4) - 3)/(-3(-1(4x + 5)^2 + 4(4x + 5) + 4) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(1|(2sin(2πx) + 1) + 2| - 1)^2 + 3(1|(2sin(2πx) + 1) + 2| - 1) + 4 and g(x) = (-3(2cos(2πx + 2) - 2) + 2)/(2(2cos(2πx + 2) - 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(4(2|(-3x + 5) - 3| - 2)^2 - 1(2|(-3x + 5) - 3| - 2) + 3) - 1)/(-1(4(2|(-3x + 5) - 3| - 2)^2 - 1(2|(-3x + 5) - 3| - 2) + 3) - 3) and g(x) = -1(-2sin(1π(4x - 1) + 1) + 1)^2 + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in L... | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-2e^(2(1x)) + 1))/(-1(-2e^(2(1x)) + 1) + 2) and g(x) = -3sin(1π((-1x - 1)/(3x + 1))) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "How many solutions does the equation -3sin(2π((-2(4x^2 - 1x - 4) + 3)/(1(4x^2 - 1x - 4))) + 1) - 3 = -4(-2sin(1πx) - 1) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 57 | MATH |
[
{
"content": "How many solutions does the equation 2cos(2π(-1(1(1|x - 1| - 2)^2 + 5(1|x - 1| - 2) - 2) - 5) - 3) - 3 = 1((-2(1sin(3πx - 1) + 1) - 3)/(-2(1sin(3πx - 1) + 1) + 3)) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 62 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1sin(3π(1(-1|x - 2| - 2)^2 + 1(-1|x - 2| - 2) - 1) - 3) + 1 and g(x) = (2(-1x^2 - 1) - 3)/(2(-1x^2 - 1) - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 804 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1((-2(-2sin(2πx) - 3) + 2)/(3(-2sin(2πx) - 3) + 2)) and g(x) = -1e^(2((-3x - 1)/(3x + 3))) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(-2(-2|(1sin(3πx) + 2) - 1| + 2)^2 - 3(-2|(1sin(3πx) + 2) - 1| + 2) + 4) - 4 and g(x) = -1((3(5x - 1) + 1)/(-1(5x - 1) + 2))^2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 62 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-2(-5(-3|x - 1| + 1)^2 - 1(-3|x - 1| + 1) + 2) - 1) + 3)/(1(-2(-5(-3|x - 1| + 1)^2 - 1(-3|x - 1| + 1) + 2) - 1)) and g(x) = 1(2|(-3x - 1)| - 3)^2 - 2(2|(-3x - 1)| - 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \... | 7 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(5(-3cos(2πx + 3) - 3) + 1) - 2| - 1 and g(x) = 3|(-1cos(1πx - 1) - 3) - 1| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-2sin(3π(3|x - 3| + 2) + 1) - 1) and g(x) = -1cos(1π(5x^2 - 2x + 5) - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 180 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(-2sin(1π(-2x - 4) - 2))^2 + 4(-2sin(1π(-2x - 4) - 2)) - 4 and g(x) = 1|(-1x - 4) + 2| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 64 | MATH |
[
{
"content": "How many solutions does the equation -2|(4((-2(3cos(1πx) + 2) + 3)/(-3(3cos(1πx) + 2) - 3)) - 1) + 1| - 1 = -1(-1|(2cos(3πx) + 3) + 2| - 2)^2 - 1(-1|(2cos(3πx) + 3) + 2| - 2) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(3cos(2π((-3(-4x) - 2)/(-1(-4x) + 3)) - 1) - 2)^2 + 4(3cos(2π((-3(-4x) - 2)/(-1(-4x) + 3)) - 1) - 2) - 5 and g(x) = 1|(-3(-3sin(1πx - 3) - 3) + 1) - 3| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your... | 97 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1e^(-1(4((3(-3|x - 3| - 2) + 3)/(1(-3|x - 3| - 2))) - 3)) + 1 and g(x) = 3|(-2cos(2π(-1x + 1) + 1) - 1) - 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "How many solutions does the equation 2(1e^(2((3(1x + 1) - 3)/(3(1x + 1) - 2))) - 3)^2 - 2(1e^(2((3(1x + 1) - 3)/(3(1x + 1) - 2))) - 3) - 4 = 1sin(2π(-5(3|x + 3|) + 4) - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 12 | MATH |
[
{
"content": "How many solutions does the equation -2((-2(1(1|x + 2|) - 5) + 1)/(1(1(1|x + 2|) - 5) + 1))^2 - 2((-2(1(1|x + 2|) - 5) + 1)/(1(1(1|x + 2|) - 5) + 1)) - 2 = -3|(1(3sin(3πx - 3) - 3) + 1) + 3| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 48 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-1|(-5(3cos(3πx))) + 3|))/(-3(-1|(-5(3cos(3πx))) + 3|) - 3) and g(x) = 5(3sin(1π((-1x + 3)/(1x + 1)) - 3) - 1)^2 - 4(3sin(1π((-1x + 3)/(1x + 1)) - 3) - 1) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\b... | 101 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3sin(3π(-1|((-2(4x - 3) + 1)/(1(4x - 3) + 3)) - 3| + 3)) - 2 and g(x) = (3(2cos(2π(-1|x - 3|) - 1) + 1) - 1)/(-1(2cos(2π(-1|x - 3|) - 1) + 1) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 46 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(3sin(1π(-3|(2x + 3) + 1| - 1) - 1) - 2)^2 - 4(3sin(1π(-3|(2x + 3) + 1| - 1) - 1) - 2) - 1 and g(x) = 1(1|((-3x + 1)/(1x + 3)) - 1| - 1)^2 - 2(1|((-3x + 1)/(1x + 3)) - 1| - 1) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer i... | 108 | MATH |
[
{
"content": "How many solutions does the equation 5((1(2(-2cos(2πx + 3) - 1)^2 - 5(-2cos(2πx + 3) - 1) + 5) + 3)/(3(2(-2cos(2πx + 3) - 1)^2 - 5(-2cos(2πx + 3) - 1) + 5) - 1)) - 1 = (3(3cos(2π(2|x + 1| + 3)) + 1) - 3)/(-1(3cos(2π(2|x + 1| + 3)) + 1) - 2) have for x in the interval [-10, 10]?\n\nPresent the answ... | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-2(-1|(-4x + 2) + 1| + 3)^2 + 1(-1|(-4x + 2) + 1| + 3) + 3) + 1)/(-1(-2(-1|(-4x + 2) + 1| + 3)^2 + 1(-1|(-4x + 2) + 1| + 3) + 3) - 2) and g(x) = -1|(-3cos(2π((3x + 2)/(2x - 1)) - 3) + 1)| - 2 intersect in the interval [-10, 10].\n\nPresent the answe... | 39 | MATH |
[
{
"content": "How many solutions does the equation 1((-2(-2|(-2sin(1πx - 2))|) + 2)/(-3(-2|(-2sin(1πx - 2))|) - 1))^2 + 4((-2(-2|(-2sin(1πx - 2))|) + 2)/(-3(-2|(-2sin(1πx - 2))|) - 1)) - 4 = (2(-1|(-2x + 2)|))/(1(-1|(-2x + 2)|) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\... | 78 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|((2(-5(-3x - 5)^2 - 1) + 1)/(-2(-5(-3x - 5)^2 - 1))) + 3| + 2 and g(x) = (3(-3(-3cos(1πx - 3) + 2) + 1) - 3)/(1(-3(-3cos(1πx - 3) + 2) + 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2((1(-3|(2cos(3πx) - 2) + 3| + 3) + 2)/(3(-3|(2cos(3πx) - 2) + 3| + 3) + 2)) and g(x) = (3(-2sin(3π(-4x^2 + 2x)) - 2) + 1)/(-2(-2sin(3π(-4x^2 + 2x)) - 2) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 694 | MATH |
[
{
"content": "How many solutions does the equation -3sin(2π(-1(-1x - 1)^2 + 4) - 1) - 1 = -2cos(2π(-1|x + 1| - 3) - 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 126 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2sin(2π(-4(1|x + 2|) - 1)) - 1 and g(x) = -3(2sin(1πx + 3))^2 - 1(2sin(1πx + 3)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 56 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(3cos(3π(1x - 1) + 1) - 3) + 1)/(2(3cos(3π(1x - 1) + 1) - 3) + 3) and g(x) = -2|((-2x - 1)/(-1x)) - 2| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(3e^(1(-5(1x + 1)^2 + 1(1x + 1) + 1))) + 1| - 1 and g(x) = 2(-1(3cos(3πx + 3))^2 + 4(3cos(3πx + 3)) + 3) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-2cos(1π(1(5x^2 - 3x - 5) - 1) - 3) - 2) + 1| - 3 and g(x) = 1(-2(2|x| - 3)^2 + 2(2|x| - 3) + 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-4((-2x + 3)/(-3x - 1))^2 - 5((-2x + 3)/(-3x - 1)) - 2)| + 2 and g(x) = 2(3|x - 3| - 3)^2 - 5(3|x - 3| - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(1|((2(3sin(1πx) + 2) + 3)/(-1(3sin(1πx) + 2))) + 3| + 1)^2 + 3 and g(x) = -1cos(3π(-3|((-1x)/(-3x)) + 1| + 1) - 2) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(2(-5(1cos(3πx)) - 2)^2 - 3(-5(1cos(3πx)) - 2) + 2) + 2)/(-3(2(-5(1cos(3πx)) - 2)^2 - 3(-5(1cos(3πx)) - 2) + 2) + 4) and g(x) = -4(-2|((3x + 3)/(1x - 2)) - 1| - 1) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxe... | 46 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1sin(1π(4((-3x - 3)/(-2x + 3))^2) + 1) + 1 and g(x) = 2((2x - 1)/(-1x + 2)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "How many solutions does the equation 1(-3(-2cos(1π(-2|x - 1| - 1) + 2) + 3) + 2)^2 - 4(-3(-2cos(1π(-2|x - 1| - 1) + 2) + 3) + 2) + 3 = 3|(-3(1x + 1)^2 + 5(1x + 1)) + 3| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(1cos(1π((-2x - 2)/(1x - 2)) + 1) - 2)^2 - 2(1cos(1π((-2x - 2)/(1x - 2)) + 1) - 2) - 5 and g(x) = (-3(2x^2 + 3) + 1)/(2(2x^2 + 3) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 72 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(3((-3(-3|x - 2| + 2) + 2)/(-2(-3|x - 2| + 2) - 3))^2 + 3((-3(-3|x - 2| + 2) + 2)/(-2(-3|x - 2| + 2) - 3)) + 1) + 2 and g(x) = 2|(-4(1cos(1πx + 3) - 2) + 5) + 1| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format:... | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(3π(5(1|x|) - 3) + 3) - 1 and g(x) = 1|(3x^2 - 2x - 1) + 2| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
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